c
Math 150, Fall 2008, Benjamin
Aurispa
Chapter 7: Analytic Trigonometry
Note: There are quite a few identities in Chapter 7. I will let you know which you need to memorize, and
which you don’t. Start learning them now. Don’t wait until right before the next test. The good thing
about all the identities is that many of them are just found by rearranging other identities.
7.1 Trigonometric Identities
Most of the following identities you already know. The new ones are the cofunction identities.
Reciprocal Identities
csc x =
1
sin x
sec x =
tan x =
sin x
cos x
1
cos x
cot x =
cot x =
1
tan x
cos x
sin x
Pythagorean Identities
sin2 x + cos2 x = 1
tan2 x + 1 = sec2 x
1 + cot2 x = csc2 x
Even-Odd Identities
sin(−x) = − sin x
cos(−x) = cos x
tan(−x) = − tan x
The following cofunction identities can easily be seen by using triangles.
Cofunction Identities
sin
π
− u = cos u
2
π
csc
− u = sec u
2
cos
π
− u = sin u
2
π
sec
− u = csc u
2
Simplify the following trig expressions.
• sin(−u) + cot(−u) cos(−u)
•
sin x cos x
+
csc x sec x
1
tan
π
− u = cot u
2
π
cot
− u = tan u
2
c
Math 150, Fall 2008, Benjamin
Aurispa
•
1 + sin u sin( π2 − u)
+
cos u
1 + sin u
Tips to Proving Trigonometric Identities
1. Start with one side of the equation and transform it to the other side of the equation. It’s usually
easier to start with the more complicated side. Do NOT just do the same operations to both
sides of the equation.
2. Use identities that you already know to change the side you started with. Combine fractions under a
common denominator (if necessary), then factor and simplify.
3. A good strategy if you are stuck is to covert everything into sines and cosines. These are the functions
you are most familiar with.
4. Always keep in mind what you are trying to get to. This will help you know what to do if you are
stuck.
Prove the following trig identities.
•
cot x sec x
=1
csc x
•
sec x + csc x
= sin x + cos x
tan x + cot x
2
c
Math 150, Fall 2008, Benjamin
Aurispa
•
1 + sin x 1 − sin x
−
= 4 tan x sec x
1 − sin x 1 + sin x
Trig Substitution is a technique you WILL use in Calculus II.
Example: Substitute x = 2 tan θ into the expression
x2
√
1
and simplify. Assume that 0 ≤ θ < π2 .
2
4+x
7.2 Addition and Subtraction Formulas
Formulas for Sine:
sin(s + t) = sin s cos t + cos s sin t
sin(s − t) = sin s cos t − cos s sin t
Formulas for Cosine:
cos(s + t) = cos s cos t − sin s sin t
cos(s − t) = cos s cos t + sin s sin t
Formulas for Tangent:
tan s + tan t
1 − tan s tan t
tan s − tan t
tan(s − t) =
1 + tan s tan t
tan(s + t) =
With these formulas, we now can evaluate sines and cosines of angles other than 30◦ , 45◦ , and 60◦ .
Find the exact value of each expression.
• sin 15◦
• cos 7π
12
3
c
Math 150, Fall 2008, Benjamin
Aurispa
• tan 75◦
Prove the cofunction identity csc( π2 − u) = sec u by using an addition or subtraction formula.
Prove the following identities.
• 1 − tan x tan y =
cos(x + y)
cos x cos y
• sin(x + y) − sin(x − y) = 2 cos x sin y
• cos(x + y) cos(x − y) = cos2 x − sin2 y
4